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anonymous

  • one year ago

I need help understanding integration by substitution: 63/(9x+2)^8

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  1. phi
    • one year ago
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    if you let u = 9x+2 what is du ?

  2. xapproachesinfinity
    • one year ago
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    This is the as saying f(x) =9x+2 What is f'(x) Just a difference of differentials but you don't need to worry about thst

  3. anonymous
    • one year ago
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    du=9 right?

  4. xapproachesinfinity
    • one year ago
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    Yes just one thing you missed Du=9dx

  5. phi
    • one year ago
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    no du/dx =9 but if you "take the derivative" u= 9x you differentiate the variables, in this case the u and the x like this du = 9 dx

  6. phi
    • one year ago
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    and once you have du = 9 dx then you also have \[ dx = \frac{1}{9} du \]

  7. phi
    • one year ago
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    now do your variable substitution replace 9x+2 with u replace dx with 1/9 du

  8. phi
    • one year ago
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    and if we have limits, we would also change the limits.

  9. phi
    • one year ago
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    and because it is easy to integrate a power, I would use the -8 exponent and get rid of the fraction

  10. anonymous
    • one year ago
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    So would it be: \[63u ^{-8}*1/9 du\] ??

  11. phi
    • one year ago
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    yes and the problem is \[ 7 \int u^{-8} du \] which you can do , right?

  12. xapproachesinfinity
    • one year ago
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    Yes

  13. phi
    • one year ago
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    once you integrate, replace u with 9x+2 to put the answer in terms of x

  14. SolomonZelman
    • one year ago
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    \(\large\color{slate}{\displaystyle\int\limits_{~}^{~}\frac{63}{(9x+2)^8}~dx}\) \(\large\color{slate}{\displaystyle u=9x+2}\) \(\large\color{slate}{\displaystyle du=(9x+2)'~\cdot dx~~~\rightarrow~~~du=9~dx}\) (you already have a 9 and dx to replace that by du, just need to re-write it. \(\large\color{slate}{\displaystyle\int\limits_{~}^{~}\frac{7\cdot 9}{(9x+2)^8}~dx}\) \(\large\color{slate}{\displaystyle\int\limits_{~}^{~}\frac{7}{(9x+2)^8}~(9\cdot dx)}\) substitution:: \(\large\color{slate}{\displaystyle\int\limits_{~}^{~}\frac{7}{(u)^8}~(du)}\) and then all you have left:: \(\large\color{slate}{\displaystyle\int\limits_{~}^{~}7~u^{-8}~du}\)

  15. SolomonZelman
    • one year ago
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    (APPLY THE POWER RULE, AND don't forget to substitute the x back for u.)

  16. SolomonZelman
    • one year ago
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    igtg....

  17. anonymous
    • one year ago
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    Ok, I got it! Thank you all!

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